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A (mathematical) Introduction to Ocean Circulation Box Models Julie Leifeld October 13, 2015

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A (mathematical) Introduction to Ocean Circulation Box Models Julie Leifeld October 13, 2015
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Julie Leifeld
A (mathematical) Introduction to Ocean
Circulation Box Models
Julie Leifeld
University of Minnesota
October 13, 2015
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Outline
Julie Leifeld
• What are Box Models?
• Stommel’s Box Model
• Cessi’s Model
• Roberts and Saha’s Model
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
What are Box Models?
(And why do we use them?)
Julie Leifeld
• What is the simplest model which can accurately describe
ocean phenomena?
• A box model divides the ocean into large “boxes”, and makes
the assumption that the water in each box is well mixed.
• This allows the mathematician to write down low dimensional
differential equations governing the behavior of the water,
which can then be studied with dynamical systems
techniques.
• The point is to look at large scale, long term behavior, as
opposed to detailed behavior, but can still give insight into
the real system.
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Julie Leifeld
Stommel’s Two Box Model
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Stommel’s Two Box Model
Julie Leifeld
kq = ρ1 − ρ2
q > 0 if the flow goes from
tank 1 to tank 2, and
q < 0 otherwise.
ρ = ρ0 (1 − αT + βS)
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Stommel’s Two Box Model
Julie Leifeld
dT1
dt
dT2
dt
dS1
dt
dS2
dt
= c(T − T1 ) − |q|T1 + |q|T2
= c(−T − T2 ) + |q|T1 − |q|T2
= d(S − S1 ) − |q|S1 + |q|S2
= d(−S − S2 ) + |q|S1 − |q|S2
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Stommel’s Two Box Model
Julie Leifeld
The symmetry of the system suggests we should look at solutions
where T1 = −T2 , and S1 = −S2 .
Let z = T1 + T2 .
dz
= cT − cT − c(T1 + T2 ) − |q|(T1 + T2 ) + |q|(T1 + T2 ) = −cz
dt
So, the T1 = −T2 is invariant and attracting.
Doing this dimensional reduction we get
dT
dt
=
c(T − T ) − 2|q|T
dS
dt
=
d(S − S) − 2|q|S
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Stommel’s Two Box Model
Julie Leifeld
Nondimensionalize!
Let τ = ct, δ = dc , y = TT , and x =
The system becomes:
dy
dτ
=
dx
dτ
= δ(1 − x) − |f |x
S
S.
1 − y − |f |y
where f is the nondimensionalized flow:
λf = −y + Rx,
f=
2q
c ,
λ=
R=
βS
αT
ck
4ρ0 αT
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Stommel’s Two Box Model
Julie Leifeld
Equilibrium solutions occur at
x =
y
=
1
|f |
1+ δ
1
1+|f |
So, we have
λf = −y + Rx = −
1
R
+
= φ(f, R, δ)
1 + |f | 1 + |f |
δ
The existence of multiple equilibria depend on λ, R, and δ. For
certain R and δ, it is possible to have three equilibrium solutions.
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Stommel’s Two Box Model
Julie Leifeld
R = 2, δ = 61 , λ = 15 .
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Cessi’s Model
Julie Leifeld
• Can changes in external forcing can cause a transition
between the stable states found in Stommel’s model?
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Cessi’s Model
Julie Leifeld
Cessi’s model is a variation on Stommel’s two box model.
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Cessi’s Model
Julie Leifeld
T˙2
1 + αS (S − S0 ) − αT (T − T0 )
= −t−1
T1 − θ2 − 12 Q(∆ρ)(T1 − T2 )
r
= −t−1
T2 + θ2 − 12 Q(∆ρ)(T2 − T1 )
r
S˙1
=
F (t)
2H S0
S˙2
=
(t)
S0 − 12 Q(∆ρ)(S2 − S1 )
− F2H
ρ/ρ0
T˙1
=
− 12 Q(∆ρ)(S1 − S2 )
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Cessi’s Model
Julie Leifeld
We again reduce the dimension of the system
(∆T = T1 − T2 , ∆S = S1 − S2 )
d∆T
dt
d∆S
dt
=
−t−1
r (∆T − θ) − Q(∆ρ)∆T
=
F (t)
H S0
Q(∆ρ) =
− Q(∆ρ)∆S
1
td
+ vq (∆ρ)2
and nondimensionalize (x =
ẋ =
ẏ =
∆T
θ
,y=
αS ∆S
αT θ ,
t = td t0 )
−α(x − 1) − x[1 + µ2 (x − y)2 ]
p(t) − y[1 + µ2 (x − y)2 ]
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Cessi’s Model
Julie Leifeld
α = td /tr is large, so this is a fast-slow system.
The slow equation:
εẋ =
ẏ
=
−(x − 1) − εx[1 + µ2 (x − y)2 ]
p(t) − y − µ2 y(1 − y)2
The fast equation:
x0
y0
= −(x − 1) − εx[1 + µ2 (x − y)2 ]
= ε(p(t) − y − µ2 y(1 − y)2 )
The (normally hyperbolic) critical manifold: x = 1.
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Cessi’s Model
Julie Leifeld
For any fixed time, we can find a potential function for behavior
on the critical manifold
4
y
2 3
y2
y2
2
+µ
− y +
− (p)y
U (y) =
2
4
3
2
0.05
0.2
-0.05
-0.10
0.4
0.6
0.8
1.0
1.2
1.4
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Julie Leifeld
Cessi’s Model
Let p be of the form p(t) = p + p0 (t) with

t≤0
 0
∆ 0≤t≤τ
p0 (t) =

0
t>τ
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Julie Leifeld
Cessi’s Model
Once the salinity forcing turns on, y changes according to the
integral
Z τ
Z y
dỹ
=
dt
2
2
0
ya −[1 + µ (ỹ − 1) ]ỹ + p + ∆
A transition between the two stable states depends on the time
over which the forcing is applied.
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Cessi’s Model
Julie Leifeld
This also shows that a critical forcing amplitude is necessary for
the transition, i.e. total volume of fresh water is not the
determining factor!
p + ∆0 =
2
2
+ µ2 (±1 + (1 − 3µ−2 )3/2 )
3 27
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Roberts and Saha
Julie Leifeld
Adding a pulse of fresh water forcing can push Stommel’s model
into different stable states. What if the salinity forcing is more
continuous?
x0
y0
µ0
=
=
=
1 − x − εA|x − y|x
ε(µ − y − A|x − y|y)
εδ(1 + ax − by)
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Roberts and Saha
Julie Leifeld
On x = 1,
ẏ =
µ̇ =
µ − y − A|1 − y|y
δ0 (λ − y)
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Julie Leifeld
Roberts and Saha
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Julie Leifeld
Dansgaard-Oeschger Events
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
Conclusions
Julie Leifeld
• Ocean box models can have important implications for
climate science.
• Ocean box models can have some really cool and complicated
math.
A (mathematical)
Introduction
to Ocean
Circulation
Box Models
References
Julie Leifeld
Stommel, Henry. ”Thermohaline convection with two stable
regimes of flow.” Tellus A 13.2 (2011).
Cessi, Paola. ”A simple box model of stochastically forced
thermohaline flow.” Journal of physical oceanography 24.9 (1994):
1911-1920.
Roberts, Andrew. ”Relaxation oscillations in an idealized ocean
circulation model.” arXiv preprint arXiv:1411.7345 (2014).
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